On Traveling Wave Solutions of the Θ−class of Dispersive Equations

We investigate traveling wave solutions to a class of dispersive models – the θ-equation of the form ut − utxx + uux = θuuxxx + (1− θ)uxuxx, including two integrable equations, the Camassa-Holm equation (θ = 1/3) and the Degasperis-Procesi equation(θ = 1/4) as special models. When 0 ≤ θ ≤ 1 2 , strong solutions of the θ-equation may blow-up in finite time, correspondingly, the traveling wave solutions are quite rich. It is shown that when θ = 0, only periodic travel wave is permissible, and when θ = 1/2 traveling waves may be solitary, periodic or kink-like waves. For 0 < θ < 1/2, traveling waves such as periodic, solitary, peakon, peaked periodic, cusped periodic, or cusped soliton are all permissible.